Question: Evaluate $\sqrt{2 -\!\sqrt{2 - \!\sqrt{2 - \!\sqrt{2 - \cdots}}}}$.
Answer: Letting $x= \!\sqrt{2 - \!\sqrt{2 - \!\sqrt{2 - \!\sqrt{2 - \cdots}}}}$, we have $x = \!\sqrt{2 - x}$.  Squaring both sides gives $x^2 = 2 - x$, so $x^2 + x -2 = 0$.  Factoring the left side gives $(x+2)(x-1) = 0$. Therefore, $x=-2$ or $x=1$.  Clearly $x$ must be positive, so we have $x= \boxed{1}$.